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Equational type logic is an extension of (conditional) equational logic, that enables one to deal in a single, unified framework with diverse phenomena such as partiality, type polymorphism and dependent types. In this logic, terms may denote types as well as elements, and atomic formulae are either equations or type assignments. Models of this logic are type algebras, viz. universal algebras equipped with a binary relation-to support type assignment. Equational type logic has a sound and complete calculus, and initial models exist. The use of equational type logic is illustrated by means of simple examples, where all of the aforementioned phenomena occur. Formal notions of reduction and extension are introduced, and their relationship to free constructions is investigated. Computational aspects of equational type logic are investigated in the framework of conditional term rewriting systems, genera!izing known results on confluence of these systems. Finally, some closely related work is reviewed and future research directions are outlined in the conclusions. * 41, Ird.wy'~~-rted (conditional) equational logic is the most established basis to the algebraic approach to abstract data type (ADT) specification [ 15,9]. algebras [ 17,4] are the standard models of this lo&c, extending u structures in a straightforward way. In algebraic specification, however, several phenomena indicate that this logic encounters limitations in practice. We mention a few, most interesting of these phenomena (which are discussed in Section 2): partiaiity, exception handling, extension, type polymorphism, dependent types. Several formal frameworks have been designed to solve the problems that are raised by e&r& of these phenomena. In particular, many of these frameworks are based on extensions of equational logic in various forms. Most of these approaches address the phenomena of their interest at a rather high level of generality. Yet, a unifying approach, where all of these phenomena can be dealt with, does not seem to have emerged. The following problem is addressed in this paper: to nd and investigate a parsimonious logic of types where al2 of the aforementioned phenomena can be dealt with in an algebraic setting. In Section 2 we further motivate our irwstigation ng, for each of those phenomena, rt discussion an a simple exam@ational type logic

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In recent years, the analysis of genomes by means of strings of length k occurring in the genomes, called k-mers, has provided important insights into the basic mechanisms and design principles of genome structures. In the present study, we focus on the proper choice of the value of k for applying information theoretic concepts that express intrinsic aspects of genomes. The value k = lg2(n), where n is the genome length, is determined to be the best choice in the definition of some genomic informational indexes that are studied and computed for seventy genomes. These indexes, which are based on information entropies and on suitable comparisons with random genomes, suggest five informational laws, to which all of the considered genomes obey. Moreover, an informational genome complexity measure is proposed, which is a generalized logistic map that balances entropic and anti-entropic components of genomes and is related to their evolutionary dynamics. Finally, applications to computational synthetic biology are briefly outlined.

Some computational aspects and behavioral patterns of P systems are considered, emphasizing dynamical properties that turn useful in characterizing the behavior of biological and biochemical systems. A framework called state transition dynamics is outlined in which general dynamical concepts are formulated in completely discrete terms. A metabolic algorithm is defined which computes the evolution of P systems modeling important phenomena of biological interest once provided with the information on the initial state and reactivity parameters, or growing factors. Relationships existing between P systems and discrete linear systems are investigated. Finally, exploratory considerations are addressed about the possible use of P systems in characterizing the oscillatory behavior of biological regulatory networks described by metabolic graphs

Some recent types of membrane systems have shown their potential in the modelling of specific processes governing biological cell behavior. These models represent the cell as a huge and complex dynamical system in which quantitative aspects, such as biochemical concentrations, must be related to the discrete informational nature of the DNA and to the function of the organelles living in the cytosol. In an effort to compute dynamical (especially oscillatory) phenomena—so far mostly treated using differential mathematical tools—by means of rewriting rules, we have enriched a known family of membrane systems (P systems), with rules that are applied proportionally to the values expressed by real functions called reaction maps. Such maps are designed to model the dynamic behavior of a biochemical phenomenon and their formalization is best worked out inside a family of P systems called PB systems. The overall rule activity is controlled by an algorithm that guarantees the system to evolve consistently with the available resources (i.e., objects). Though radically different, PB systems with reaction maps exhibit very interesting, often similar dynamic behavior as compared to systems of differential equations. Successful simulations of the Lotka-Volterra population dynamics, the Brusselator, and the Protein Kinase C activation foster potential applications of these systems in computational systems biology.

Accessibility of various tools is listed in methods; the Log-Gain Stoichiometric Stepwise algorithm is accessible at http://www.cbmc.it/software/Software.php.